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YCor
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Let $G$ be a topological group andand $X$ be a topological space. Assume thatLet $\alpha, \beta:G\times X\to X$$\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions conectingconnecting $\alpha$ to $\beta$. This means that there is a continuoscontinuous function $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

Question 1: What is an example of the following situation: There are two group actions $\alpha, \beta$$\alpha$, $\beta$ by $ G=S^1$$G=S^1$ on a manifold  $M$ which are not homotopic actions but for every $g\in G$, the two homeomirphismshomeomorphisms $\alpha_g, \beta_g$$\alpha_g$, $\beta_g$ of $M$ are homotopic mapmaps?

Question 2: What is an example of the following situation: There are two group actions $\alpha, \beta$$\alpha$, $\beta$ by $ G=S^1$ on a manifold  $M$ which are not homotopic actions but $\alpha, \beta$ are homotopic maps as maps from $G\times X$ to $X$? Ý

Let $G$ be a topological group and $X$ be a topological space. Assume that $\alpha, \beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions conecting $\alpha$ to $\beta$. This means that there is a continuos function $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

Question 1: What is an example of the following situation: There are two group actions $\alpha, \beta$ by $ G=S^1$ on a manifold  $M$ which are not homotopic actions but for every $g\in G$ two homeomirphisms $\alpha_g, \beta_g$ of $M$ are homotopic map?

Question 2: What is an example of the following situation: There are two group actions $\alpha, \beta$ by $ G=S^1$ on a manifold  $M$ which are not homotopic actions but $\alpha, \beta$ are homotopic maps as maps from $G\times X$ to $X$? Ý

Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions connecting $\alpha$ to $\beta$. This means that there is a continuous function $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

Question 1: What is an example of the following situation: There are two group actions $\alpha$, $\beta$ by $G=S^1$ on a manifold $M$ which are not homotopic actions but for every $g\in G$, the two homeomorphisms $\alpha_g$, $\beta_g$ of $M$ are homotopic maps?

Question 2: What is an example of the following situation: There are two group actions $\alpha$, $\beta$ by $ G=S^1$ on a manifold $M$ which are not homotopic actions but $\alpha, \beta$ are homotopic maps as maps from $G\times X$ to $X$?

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Ali Taghavi
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Let $G$ be a topological group and $X$ be a topological space. Assume that $\alpha, \beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions conecting $\alpha$ to $\beta$. This means that there is a continuos mapfunction $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

Question 1: What is an example of the following situation: There are two group actions $\alpha, \beta$ by G=S^1$$ G=S^1$ on a manifold $M$ which are not homotopic actions but for every $g\in G$ two homeomirphisms $\alpha_g, \beta_g$ of $M$ are homotopic map?

Question 2: What is an example of the following situation: There are two group actions $\alpha, \beta$ by $ G=S^1$ on a manifold $M$ which are not homotopic actions but $\alpha, \beta$ are homotopic maps as maps from $G\times X$ to $X$? Ý

Let $G$ be a topological group and $X$ be a topological space. Assume that $\alpha, \beta:G\times X\to X$ be two group actions. We say these two actions are homotopic actions if there is a continuous path of actions conecting $\alpha$ to $\beta$. This means that there is a continuos map $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

What is an example of following situation: There are two group actions $\alpha, \beta$ by G=S^1$ on a manifold

Let $G$ be a topological group and $X$ be a topological space. Assume that $\alpha, \beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions conecting $\alpha$ to $\beta$. This means that there is a continuos function $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

Question 1: What is an example of the following situation: There are two group actions $\alpha, \beta$ by $ G=S^1$ on a manifold $M$ which are not homotopic actions but for every $g\in G$ two homeomirphisms $\alpha_g, \beta_g$ of $M$ are homotopic map?

Question 2: What is an example of the following situation: There are two group actions $\alpha, \beta$ by $ G=S^1$ on a manifold $M$ which are not homotopic actions but $\alpha, \beta$ are homotopic maps as maps from $G\times X$ to $X$? Ý

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Ali Taghavi
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Homotopy of group actions

Let $G$ be a topological group and $X$ be a topological space. Assume that $\alpha, \beta:G\times X\to X$ be two group actions. We say these two actions are homotopic actions if there is a continuous path of actions conecting $\alpha$ to $\beta$. This means that there is a continuos map $\Gamma:[0,1]\times G\times X \to X$ such that each $\Gamma_t$ is a group action and we have $\Gamma_0=\alpha, \Gamma_1=\beta$.

What is an example of following situation: There are two group actions $\alpha, \beta$ by G=S^1$ on a manifold